All Questions
Tagged with tensor-calculus quantum-field-theory
30
questions
0
votes
0
answers
38
views
Bitensors at three or more space-time points
Bitensors, i.e. tensors at two points that have indices belonging to either of them, have been used in the literature quite a bit and there are many calculations involving them. They are the go-to ...
0
votes
1
answer
83
views
How can I calculate the square of Pauli-Lubanski vector in a rest frame?
recently I've been trying to demonstrate that, $$\textbf{W}^2 = -m^2\textbf{S}^2$$ in a rest frame, with $W_{\mu}$ defined as $$W_{\mu} = \dfrac{1}{2}\varepsilon_{\mu\alpha\beta\gamma}M^{\alpha\beta}p^...
- 1
2
votes
1
answer
106
views
Primary operators in $d=3$ (bosonic free) conformal field theory
Consider the free bosonic conformal field theory (CFT) in spacetime dimension $d=3$. I would like to explicitly construct a primary operator of spin $l=4$, with four scalar fields $\phi$ and five ...
- 165
2
votes
0
answers
22
views
Dimensional regularisation and Wick theorem [duplicate]
Consider an integral:
$$I^{ij}=\int\frac{d^d\textbf{p}}{(2\pi)^d}p^i p^j f(\textbf{p}^2).$$
How can we show that this is equal to:
$$I^{ij}=\frac{\delta^{ij}}{d}\int\frac{d^d\textbf{p}}{(2\pi)^d}\...
- 183
1
vote
1
answer
103
views
Suppressed indices - when and why?
When following textbooks in QFT I have many times seen the usage of 'suppressed indices'. This is a bit confusing to me, sometimes it is done without any explanation. For instance I usually see that ...
- 311
1
vote
0
answers
67
views
Correlation function of 4-currents on a general QFT
Given $V^\mu(x)$ a 4-current of a general unitary, Poincaré invariant QFT, I need to show that the correlation function:
$$iC_{\mu\nu}(x-y) = \langle {\tilde{0}}|T\left[V_\mu(x)V_{\nu}^\dagger(y)\...
- 21
3
votes
1
answer
2k
views
How to use Passarino-Veltman reduction for integral containing two tensors?
I am trying to use Passarino-Veltman reduction to solve the following integral:
$$ \int \frac{d^{D}k}{(2\pi)^{D}} \frac{k^{\mu}k^{\nu}}{k^{2}(k-q)^{2}} $$
However every ansatz I try, the integral goes ...
- 115
1
vote
0
answers
57
views
Interpreting notation of tensors in QFT [duplicate]
I am having a really hard time wrapping my head around component notation for tensor fields. For example, I do not know exactly what the following expression means
$$\partial_\mu\partial^\nu \phi, \...
- 133
0
votes
1
answer
280
views
Matrix "dimensional analysis" of Lagrangians in QFT
Since the important things in the QFT Lagrangian are vectors and matrices, I wanted to do a "matrix dimensional analysis" of each term.
The electromagnetic Lagrangian (ignoring all constants ...
- 332
1
vote
1
answer
84
views
Is the off-diagonal part of this rank-2 tensor integrand odd?
Peskin and Schroeder in Introduction to Quantum Field Theory consider the following tensor integral (Eq. 6.46):
$$\int \frac{\mathrm{d}^4l}{(2\pi)^4} \frac{l^\mu l^\nu}{D^n} = \int \frac{\mathrm{d}^4 ...
- 155
0
votes
1
answer
64
views
Question solving tensor problems for the Special Conformal Killing Equation
Background
I know that following index notation, these are true:
$$\partial_\mu x^\nu = \delta^\mu _\nu \hspace{5mm} and \hspace{5mm} \partial_\mu x_\nu = \eta_{\mu\nu} \tag{1}$$
Exercise
Knowing ...
- 753
1
vote
1
answer
272
views
Quantization of photon field
I was following Prof David Tong's notes on quantum field theory (Chapter 6, page 134). Consider a physical state $|\Psi\rangle$ in Fock space for N photons. If we consider these photons to have 4 ...
- 369
1
vote
1
answer
134
views
Momentum operator in spacelike metric convention
What are the underlying principles for choosing signs for the momentum operators in QM/QFT?
Let's say, for the $(+,-,-,-)$ metric convention we have $\partial^\mu = (+\partial_0,-\nabla)$. Why not $\...
1
vote
4
answers
432
views
Is this useful identity valid only under the integral sign?
Studying dimensional reugularization one often encounters the following identity:
$$
\int d^d q\, \, q^\mu q^\nu f(q^2) = \frac{1}{d}g^{\mu\nu}\int d^d q\,\, q^2 f(q^2)
$$
often justified by some ...
- 505
6
votes
1
answer
674
views
The spinor metric, basic spinor calculations and spinor indices
I'm currently reading the textbook "Finite Quantum Electrodynamics" by Günter Scharf, but I find myself stuck already on page 24.
Background
Scharf introduces the index-raising symbol (spinor metric)...
- 144